Let $\mathcal{C}$ be a category which admits inductive limits. One says that an object $X$ of $\mathcal{C}$ is of finite type if for any functor $\alpha: I\to \mathcal{C}$ with $I$ a direct set, the natural map $$\lim\limits_{\to}\text{Hom}_{\mathcal{C}}(X,\alpha) \to\text{Hom}_{\mathcal{C}}(X,\lim\limits_{\to}\alpha)$$ is injective. Show that this definition coincides with the classical one when $\mathcal{C}=\text{Mod}(R)$ is the category of $R$-modules for a ring $R$.
I have no idea about this problem. Could you please give me any hints? Thanks.